Clase 1 - Control En Matlab

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CONTROL 2 Uso del Matlab Considere el siguiente sistema:

1 0   x1  0  x&1   0  x&  =  0 0 1   x2  + 0  u 2         x&3  − 6 − 11 − 6  x3  1  x1  y = [1 1 1] x2     x3  Lo primero es digitar las matrices en Matlab a=[0,1,0;0,0,1;-6,-11,-6] a= 0 1 0 0 0 1 -6 -11 -6 b=[0;0;1] b= 0 0 1 c=[1,1,1] c= 1

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d=[0] d= 0 Con el comando eig se pueden encontrar los autovalores y los autovectores help eig EIG Eigenvalues and eigenvectors. E = EIG(X) is a vector containing the eigenvalues of a square matrix X. [V,D] = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D. [V,D] = EIG(X,'nobalance') performs the computation with balancing disabled, which sometimes gives more accurate results for certain problems with unusual scaling. E = EIG(A,B) is a vector containing the generalized eigenvalues of square matrices A and B. [V,D] = EIG(A,B) produces a diagonal matrix D of generalized eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that A*V = B*V*D. See also CONDEIG, EIGS. Overloaded methods help sym/eig.m help lti/eig.m eig(a) ans = -1.0000 -2.0000 -3.0000

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[m,ed]=eig(a) m= -0.5774 0.2182 -0.1048 0.5774 -0.4364 0.3145 -0.5774 0.8729 -0.9435 ed = -1.0000 0 0 0 -2.0000 0 0 0 -3.0000 Se observa la matriz modal m y en la matriz ed los polos en la diagonal principal. Ahora se usa la orden ss help ss SS Create state-space model or convert LTI model to state space. Creation: SYS = SS(A,B,C,D) creates a continuous-time state-space (SS) model SYS with matrices A,B,C,D. The output SYS is a SS object. You can set D=0 to mean the zero matrix of appropriate dimensions. SYS = SS(A,B,C,D,Ts) creates a discrete-time SS model with sample time Ts (set Ts=-1 if the sample time is undetermined). SYS = SS creates an empty SS object. SYS = SS(D) specifies a static gain matrix D. In all syntax above, the input list can be followed by pairs 'PropertyName1', PropertyValue1, ... that set the various properties of SS models (type LTIPROPS for details). To make SYS inherit all its LTI properties from an existing LTI model REFSYS, use the syntax SYS = SS(A,B,C,D,REFSYS). Arrays of state-space models: You can create arrays of state-space models by using ND arrays for A,B,C,D above. The first two dimensions of A,B,C,D determine the number of states, inputs, and outputs, while the remaining

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dimensions specify the array sizes. For example, if A,B,C,D are 4D arrays and their last two dimensions have lengths 2 and 5, then SYS = SS(A,B,C,D) creates the 2-by-5 array of SS models SYS(:,:,k,m) = SS(A(:,:,k,m),...,D(:,:,k,m)), k=1:2, m=1:5. All models in the resulting SS array share the same number of outputs, inputs, and states. SYS = SS(ZEROS([NY NU S1...Sk])) pre-allocates space for an SS array with NY outputs, NU inputs, and array sizes [S1...Sk]. Conversion: SYS = SS(SYS) converts an arbitrary LTI model SYS to state space, i.e., computes a state-space realization of SYS. SYS = SS(SYS,'min') computes a minimal realization of SYS. See also LTIMODELS, DSS, RSS, DRSS, SSDATA, LTIPROPS, TF, ZPK, FRD. sys=ss(a,b,c,d) a= x1 x2 x3

x1 0 0 -6

x1 x2 x3

u1 0 0 1

y1

x1 1

x2 1 0 -11

x3 0 1 -6

x2 1

x3 1

b=

c=

5

d= y1

u1 0

Continuous-time model. Se usa la orden canon help canon CANON Canonical state-space realizations. CSYS = CANON(SYS,TYPE) computes a canonical state-space realization CSYS of the LTI model SYS. The string TYPE selects the type of canonical form: 'modal' : Modal canonical form where the system eigenvalues appear on the diagonal. The state matrix A must be diagonalizable. 'companion': Companion canonical form where the characteristic polynomial appears in the right column. [CSYS,T] = CANON(SYS,TYPE) also returns the state transformation matrix T relating the new state vector z to the old state vector x by z = Tx. This syntax is only meaningful when SYS is a state-space model. The modal form is useful for determining the relative controllability of the system modes. Note: the companion form is illconditioned and should be avoided if possible. See also SS2SS, CTRB, CTRBF, SS. Overloaded methods help ss/canon.m help lti/canon.m help frd/canon.m csys=canon(sys)

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a= x1 x2 x3

x1 -1 0 0

x1 x2 x3

u1 -0.86603 -4.5826 -4.7697

y1

x1 -0.57735

y1

u1 0

x2 0 -2 0

x3 0 0 -3

b=

c= x2 x3 0.65465 -0.7338

d=

Continuous-time model. El sistema quedo diagonalizado. Se observa que es completamente controlable y completamente observable. Para comprobar controlabilidad help ctrb CTRB Compute the controllability matrix. CO = CTRB(A,B) returns the controllability matrix [B AB A^2B ...]. CO = CTRB(SYS) returns the controllability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to CTRB(sys.a,sys.b).

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For ND arrays of state-space models SYS, CO is an array with N+2 dimensions where CO(:,:,j1,...,jN) contains the controllability matrix of the state-space model SYS(:,:,j1,...,jN). See also CTRBF, SS. Overloaded methods help lti/ctrb.m co=ctrb(a,b) co = 0 0 1 0 1 -6 1 -6 25 rank(co) ans = 3 El rango de la matriz de controlabilidad es 3 y es igual a la dimensión de la matriz a, por lo tanto el sistema es completamente controlable. Para verificar la observabilidad se usa el comando obsv help obsv OBSV Compute the observability matrix. OB = OBSV(A,C) returns the observability matrix [C; CA; CA^2 ...] CO = OBSV(SYS) returns the observability matrix of the state-space model SYS with realization (A,B,C,D). This is equivalent to OBSV(sys.a,sys.c). For ND arrays of state-space models SYS, OB is an array with N+2 dimensions where OB(:,:,j1,...,jN) contains the observability matrix of the state-space model SYS(:,:,j1,...,jN).

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See also OBSVF, SS. Overloaded methods help lti/obsv.m ob=obsv(a,c) ob = 1 1 1 -6 -10 -5 30 49 20 rank(ob) ans = 3 El rango de ob es igual a la diemensión de la matriz a por lo tanto el sistema es completamente observable. Veamos la respuesta en el tiempo: step(a,b,c,d) Step Response From: U(1) 0.18 0.16 0.14

To: Y(1)

Amplitude

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5

2

Time (sec.)

2.5

3

3.5

4

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[num,den]=ss2tf(a,b,c,d) num = 0 1.0000 1.0000 1.0000 den = 1.0000 6.0000 11.0000 6.0000 step(num,den) Step Response From: U(1) 0.18 0.16 0.14

To: Y(1)

Amplitude

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.5

1

1.5

2

Time (sec.)

sistema=tf(num,den) Transfer function: s^2 + s + 1 ---------------------s^3 + 6 s^2 + 11 s + 6 roots(den) ans = -3.0000 -2.0000 -1.0000

2.5

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3.5

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Simulink simulink

K Step

b

1 s Integrator

K c

Scope

a K

x' = Ax+Bu y = Cx+Du Step

State-Space

Scope